Pascal's wager is an argument based on probability theory for why one should live as if God exists, even though this cannot be proved or disproved through reason. It was formulated by Blaise Pascal.[1]

Formulation

Pascal's original formulation of the wager was written down as a fairly short paragraph in Pensées amongst several other notes that could be considered "wagers".[1] Its argument is rooted in game theory and that the best course of action is to believe in God regardless, because that option gives the biggest potential gain regardless if God is real or not. Pascal's original text is longwinded and somewhat convoluted philosophy-speak,[2] but it can be distilled more simply:

If you believe in God and God does exist, you will be rewarded with eternal life in heaven; thus an infinite gain.

If you do not believe in God and God does exist, you will be condemned to remain in hell forever; thus an infinite loss.

If you believe in God and God does not exist, you will not be rewarded; thus a finite loss.

If you do not believe in God and God does not exist, you will not be rewarded, but you have lived your own life; thus a finite gain.

The gains and losses associated to outcomes 3 and 4 can be thought of as the opportunity costs of feigning belief and living in accordance with religious norms, since these are typically more restrictive than secular laws. These costs are finite because of human mortality. Mathematically, a finite gain or loss is negligible compared to an infinite gain or loss (as would be incurred during an eternal afterlife). Therefore, he concluded that it was a much better choice to believe in God rather than not. The Wager can also be seen in table form and it becomes clear that belief gives you a reward or (practically) nothing, while disbelief gives you punishment or nothing:

God exists

God does not exist

Believe in God

Infinite gain in heaven

Insignificant loss

Disbelieve in God

Infinite loss in hell

Insignificant gain

Criticism

There are many problems with Pascal's Wager. Like most theological arguments it seems more about reassuring existing believers than converting non-believers. Taken to its logical conclusion there are places where the theological implications are undesirable as well as objections to its actually reasoning. One, which does not constitute a logical fallacy in its own right, is the fact that there are an infinite number of hypothetical gods.

An illustrative story

Alkanes are a group of hydrocarbons which have as many hydrogen atoms as possible, i.e. they are saturated. Alkanes include methane with one carbon atom), ethane with two, propane with three, butane with four, and from there being named after numerical prefixes indicating the number of carbon atoms in them (pentane, hexane, heptane, et cetera). Let us propose an infinity of hypothetical gods known as "Alkahs" (portmanteau of "alkane" and "Allah"). Thus, there would be a methane god, an ethane god, propane god, et cetera. Each of these alkahs has a monotheistic religion, and each of their doctrines are as follows:

I am the only true god.

I created the Universe from my holy alkane (the methane god's holy alkane is methane, the ethane god's is ethane, etc).

If you believe I exist, then you gain eternal paradise after death.

If you don't, you will suffer for eternity after death.

Since there are an infinity of hypothetical alkanes, each with one more carbon atom than the next, and each god is no more likely to exist than any other, there is no reason to favor one over the other. A common defense is that if you choose a religion, at least you have a non-zero chance of being correct. Not only is this false because one divided by infinity is zero (1 ÷ 2 = 0.5, 1 ÷ 3 = 0.33, tending towards zero), but we can resolve this by adding another hypothetical god, "Noewah" (combination of "no way" and "Yahweh"), whose doctrine is:

I am the only true god.

If you do not believe I exist, you gain eternal paradise after death.

If you do believe I exist, you suffer eternally after dying.

Since no god is more likely to exist than any other, placing your bets in atheism is not any more harmful than placing your bets on a particular god. Of course, due to the fact that 1/∞ = 0, this doesn't matter anyway, since you're guaranteed eternal torture. The reason you should place your bets on atheism anyway is because since everything will almost surely lead to hell, you might as well not waste your time, money, energy, and resources worshiping and performing other religious acts. By realizing that there is not only one conceivable god, Pascal's wager actually says to wager there is no god.

Of course, this analogy fails to account for gods who aren't total assholes, i.e. care more about how good a person you are than whether you believe he exists or not. This subject is covered by the Atheist's Wager.

Assumptions

In addition, Pascal's wager also makes a number of assumptions. If any of these can be shown to either be false or undesirable, then the power of the Wager for determining one's actions and beliefs is severely weakened. These mostly stem from the theological implications of applying the Wager to belief in God, rather than the Game Theory attributes and decision making process presented.

Legitimacy of worship

The Wager assumes that God will be impressed by, and happily reward, people who worship just to avoid Hell. It's not clear why a perfect creator being would want such a thing, the constant harangues and demands for worship by the Abrahamic god nothwithstanding. An all-powerful (or very powerful) being would gain little from the mental allegiance of human beings, in the same way that as a human, persuading all the inhabitants of an anthill to worship you would be pretty pointless.

If the aim is to bring about good in the world, why would a god not promise to reward everyone who does good, regardless of whether they worship it or not? What about someone who gives away their money to the poor while loudly condemning priests? There's at least a theoretical conflict between devoting all your energy to praising and worshipping god and doing good things for other people on earth, raising the question of what exactly a god would want from us anyway - actual love and obedience, or obedience out of fear?

The satirical fantasy writer Terry Pratchett had a version of Pascal's Wager in his book titled Small Gods: "Upon his death, the philosopher in question found himself surrounded by a group of angry gods with clubs. The last thing he heard was 'We're going to show you how we deal with Mister Clever Dick around here...'"

Total self-interest

The Wager also assumes that the believer will only care about maximizing their own gains. For example, it's not hard to imagine a situation where someone intentionally sets themself up for a fall in order to redeem two or more people. Hey, Judas, how's it hanging?

More troubling than this are occasions where you might theoretically be called upon to hurt someone else to advance your worship of the superior entity. In the Old Testament there are numerous instances when worshipers had to kill and hurt others as commanded by God -- in fact, there are occasions in which God was extremely displeased that they didn't take the abuse of fellow humans far enough. Even with the Pascal's Wager metric in place, one could argue that it's more moral to resist these commands for the sake of others even if it results in an infinite loss for you.

Correctness of worship

Many sects of God require more conditions for proper worship than others; some of them are downright contradictory. But if you don't do enough of the rituals required for proper worship you will still go to hell despite the ones you did perform. Consider if God decided that the Mormon way of worshiping him was correct and that the Catholic way was incorrect. Behaving at random would give you better odds (1 in an infinitesimal chance; maybe some crank in Ancient China performed the proper rituals that pleased Mormon God despite not even knowing that He existed) at salvation than being saved. Indeed, since humans have no way of knowing which way is pleasing to God you may as well not even bother; a tiny gain plus going to Hell is still better than a tiny loss plus still going to Hell.

Infinite utility

The wager assumes that going to hell will be infinitely miserable, while going to heaven would be infinitely pleasurable. Similarly, the wager also assumes that believing in a particular god is costless to the believer.

In some traditional views of heaven and hell, this reward lasts over an infinite time, thus it can be realized by a finite, even small, reward per unit of time. The present value of an infinite reward can be very small: Consider the present value of a "million dollar" lottery which pays a dollar a year for a million years; Now consider the present value of an "infinite dollar" lottery which pays one cent a year for infinity. Even if you live forever, you don't gain a lot.

Ability to "believe"

The wager assumes that a human being has the ability to believe something by an act of will: not just to say one believes it but to actually, sincerely, believe it to be true. This is known as doxastic voluntarism; it is probable that most people lack the ability to do this deliberately.

Consider how you would respond if someone told you to believe that the earth rested upon the back of a giant turtle. Even if you were inclined to, it is doubtful whether you really could genuinely believe it.

Judeo-Christian bias

Another common critique is that the wager only deals with the traditional Judeo-Christian image of a god who rewards his followers. One could imagine that the truth is in fact a perverse god who would damn his supporters and reward unbelievers simply for the sake of irony,[3] or the Maltheistic idea that "gods" are spiritual entities who eat their worshippers' souls after their deaths.[4]

Also, the wager only allows for one true god. Should there be more than one (such as in the Greek and Roman mythologies), a follower of Pascal may offend a powerful god by only worshipping a weaker one, leading to his damnation despite his worship.

Christian afterlife is possible

Some people consider parts of this scenario to be logically contradictory or incompatible with reality, which leads them to assign a probability of zero to the "Christian hell" outcome.

This may be because they believe that the Christian god is impossible or defined in a contradictory way, because they do not believe such a beneficent god could send anyone to hell, or because they believe that hell itself is impossible. This last argument may be based on the belief that pain requires a physical body, or on an argument that the limited human mind is incapable of suffering eternal torture. If you were tortured for ten times as long as you've been alive, would you be "you" anymore? How about after a million years? A trillion? At some point, whether 10 or 1010 years from now, one would expect that your mind would either break or "get used to it", probably both. Either possibility precludes the experience of infinite regret and suffering.

Unfortunately, a similar argument can be made regarding heaven. An eternal high seems impossible for a limited creature, so going to heaven would only be possible if one transformed into something not only different, but even unrecognizable to one's earthly self; in which case, what's being punished/rewarded isn't meaningfully "you" at all.

Ulterior reason for blind faith

The wager assumes that there is a self-evident reason for rewarding blind faith. Why is the faith of a believer better than the personal courage of the disbeliever that leads an outstanding life? Why does a deity prefer blind faith over evidence based submission?

The real argument goes in the opposite direction: God exists (I assume) and I cannot see him, therefore he has to have a reason not to show himself. Therefore, there has to be something for me if I play along.

Reversing the wager

One way to counter the wager is to replace Pascal's Judeo-Christian God with a perverse god that punishes those who believe in him without evidence, and rewards those who don't. Note that this doesn't even presuppose that the Bible and other holy texts are not divinely inspired: this god could have authored them to serve as a test of credulity.

God exists

God does not exist

Believe in God

Infinite loss in hell

Insignificant loss

Disbelieve in God

Infinite reward in heaven

Insignificant gain

This theoretical belief system, which is just as empirically provable as the belief system underlying Pascal's wager, presents a win/win scenario for atheists and a lose/lose scenario for those who believe in God. Since the two contrasting ideas of a specific god are logically equivalent in likelihood, atheism is shown to have the greatest potential for gain, completely negating and effectively reversing Pascal's argument.

Further reversal

It is dubious whether people can force themselves to believe whatever they want. Instead of considering whether one should believe in God, this reversal considers whether it's a good strategy for an otherwise good, socially responsible person to pretend to believe in God:

God exists

God does not exist

rewards works

rewards faith

Feigning belief

punishment

punishment

finite loss (untruthfulness to self)

Honest disbelief

reward

punishment

finite gain (honest life)

As before, feigning belief is never a better strategy than being honest about your disbelief, unless the punishment for feigned belief (trying to trick a god) is milder than the one for disbelief. True believer is left out, because you're arguing from the viewpoint that belief is something that takes place deep down inside, and is not a facade, and that you simply lack it.

Applications outside theology

While Pascal's Wager is a theologically flawed argument, the principle has applications in other aspects of life, such as investment and betting. It is a way of dealing with uncertainty by assessing the magnitude of potential gains against losses. In the Wager, believing in God produces the larger gain whereas not believing produces the greater loses. This principle of comparing potential gains and loses is a way of ensuring against black swan like events ("fragility" and "anti-fragility" according to Black Swan author Nassim Taleb). For example, placing 100% of your funds into a single investment leads to a possible loss - regardless of the odds, which may be unknowable - of 100% of your funds. Such an event would be a greater loss than if you spread your funds over many investments where a single failure (again, regardless of odds) is not devastating but your chances of success are slightly increased. The same thinking lies behind solutions to the prisoner's dilemma. Viewing unknown events through the eyes of Pascal's Wager eliminates the need to understand the probabilities behind events because decisions can be made by maximizing potential benefits and minimizing the harms. Indeed, this is the very point of the Wager if it is stripped of its theological implications.